MHB Evaluating $m^4-18m^2-8m$ Given $p,\,q,\,r$

[anemone]

Let $p,\,q,\,r$ be the roots of $x^3-9x^2+11x-1=0$ and let $m=\sqrt{p}+\sqrt{q}+\sqrt{r}$.

Evaluate $m^4-18m^2-8m$.

 

[Euge]

Re: Find m^4-18m^2-8m

Let $p,\,q,\,r$ be the roots of $x^3-9x^2+11x-1=0$ and let $m=\sqrt{p}+\sqrt{q}+\sqrt{r}$.

Evaluate $m^4-18m^2-8m$.

Spoiler

We have

$\displaystyle m^2 = p + q + r + 2(\sqrt{pq} + \sqrt{qr} + \sqrt{rp}) = 9 + 2(\sqrt{pq} + \sqrt{qr} + \sqrt{rp})$.

Thus

$\displaystyle (m^2-9)^2 = 4[pq + qr + rp + 2(\sqrt{p(pqr)} + \sqrt{q(pqr)} + \sqrt{r(pqr)})]$
$\displaystyle = 4[11 + 2(\sqrt{p} + \sqrt{q} + \sqrt{r})]$
$\displaystyle = 44 + 8m$.

Since $(m^2 - 9)^2 = m^4 - 18m^2 + 81$, we obtain $m^4 - 18m^2 + 81 = 44 + 8m$, or

$\displaystyle m^4 -18m^2 - 8m = -37$.

 

[anemone]

Awesome, Euge! Thanks for your solution and thanks for participating!:)